Файл: Antsiferov V.V., Smirnov G.I. Physics of solid-state lasers (ISBN 1898326177) (CISP, 2005)(179s).pdf

Stochastic and transition processes in solid-state lasers

As a result of the summation of the series (7.34), term-by-term differentiation and subsequent integration, the expression for the meantime to establishment of the given photon concentration, generated by the second excitation pulse, can be reduced to the following form:

here Ei(x) is the integral exponential function.

As previously, the dispersion σ 2t is expressed by means of a series. This complicates the analysis of the dependence of σ 2t on T. However, it may be shown that the function σ 2t(T) is, like the dependence of t on T, a monotonically increasing dependence. From (7.40) in the limiting case T → ∞ , we obtain

In this case, the equation for σ 2t coincides accurately with the expres-

sion (7.29), and the equation (7.41) for t corresponds to the expression (7.28) at n1 → 0. This is in agreement with the approximation of the initial distribution by the δ -function.

At T → 0, the mean time t and the dispersion σ 2t rapidly decrease. We determine the asymptotic expressions for them at N >> n2:

To calculate σ 2t in this case, we can use directly equation (7.24) where we use the first and second terms of the expansion of the modified Bessel function I0(u) into a series, and also the asymptotics of these functions at low and high values of the argument. As indicated by (7.42), (7.43), the values of t and σ 2t decrease with increase of the mean concentration of the seed photons.

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7.5 TRANSITION PROCESSES AT SLOW CHANGES OF THE LASING PARAMETERS

The excitation of lasing by pulses with a flat leading edge (in particular, trapezoidal pulses) corresponds to slow changes of the lasing parameter. The change of the lasing parameter with time in the vicinity of the threshold is assumed to be approximately linear. The process of passage of the lasing threshold in the given situation is described by the equation:

α 1 is the value of the lasing parameter below the threshold as the moment of the start of examination t = 0.

With the variation of t, the parameters α converts to 0 at the moment t = α 1/β . According to (7.28), at this moment t → ∞ . Consequently, in the vicinity of the threshold at any suitable slow change of the value of α , the transition process in the conventional meaning of the word takes place, i.e. lasing ‘does not manage to follow’ the variation of external perturbation. However, at t > α 1/β , the lasing parameter α differs from 0 and continues to increase. The time to establishment of the given photon concentration t decreases, and the lasing ‘tracks’ perturbation in a quasistationary manner:

Using the solution of equation (7.44) at (β t – α 1) >> γ n, it may be shown that the linear period of the development of lasing is determined by the time τ > 2/γ n. This time is approximately equal to the mean time of increase of lasing in the presence of fluctuations and determines the range in which the transition process takes place.

At t >> α 1/β , after the moment of time there is a steady regime of lasing with the Gaussian distribution of the photon concentration (7.9) where n is now determined by equation (7.45). The physical pattern is such that the lasing at every successive moment of time is developed on the basis of lasing in the previous moment with the approximately delta-shaped initial distribution. Naturally, the intensity of fluctuations rapidly decreases in this case. At the same time, the fluctuations of the time of decrease of lasing are considerably smaller than the fluctuations of the time of increase of lasing.

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